^{1}A complete listing of all general and special face forms of the 32 point groups is given in ch. 10 of Hahn & Klapper (2002), Table 10.1.2.2; the *eigensymmetries* of the 47 face forms are listed in Table 10.1.2.3. Illustrations of all face forms are contained in ch. 3.2 (pp. 184-188) of the book by Vainshtein (1994), ch. 3 (Fig. 73) of Shubnikov & Koptsik (1974) and ch. 10 of Buerger (1956).

^{2}Representative twin operation *m*'(10).

^{3}Note that this illustration does not require crystals to exhibit planar (habit) faces and complete face forms. It is applicable to crystals of any (also spherical) shape to be studied by X-ray diffraction. This is shown in Section 2.3, p. 336, of Klapper & Hahn (2010).

^{4}In some cases of reticular merohedral twins certain indices are *always integer*, *e.g.* for the tetragonal 5 and hexagonal 7 twins which preserve the tetragonal or hexagonal axis. Here the index *l* is *always integer* (*cf.* §§5 and 6).

^{5}The rhombohedral 3 twins treated here are `twins by reticular merohedry with *parallel* threefold axes'. They are thus independent of the *c*/*a* ratio (hexagonal axes) or the rhombohedral angle (rhombohedral axes) and can occur in any rhombohedral crystal. Twins `with *inclined* threefold axes' depend on the axial ratio or on . Both types have been derived by Grimmer (1989*b*).

^{6}They are the same as those in Table 9, subtable (c) of Klapper & Hahn (2010) for crystals with hexagonal lattices and 1 twin laws 2[001] and *m*(0001). This is due to the fact that face forms are independent of the lattice type (hexagonal or rhombohedral).

^{7}In principle, however, domains of 3 obverse/reverse reflection twins with twin laws 3 3*m*, 3 and 32 can be studied by the reversal of their optical rotation sense. The same applies to the corresponding trigonal 1 twins.

^{8}Exact lattice coincidence is, in principle, not possible because the coincidence is not enforced by symmetry as it is in the case of `parallel *c*-axis' twins. It may occur, however, for a certain temperature if the thermal expansion is anisotropic.

^{9}*Q* = (*k*_{1} - 2*P*)^{2} + *P*^{2} = (*h*_{1} - *P*)^{2}/4 + *P*^{2} for coincidence condition *h*_{1} + 2*k*_{1} = 5*P*; similar for *h*_{2}, *k*_{2} with *h*_{2} + 2*k*_{2} = 5*P*.

^{10}The *d* values of reflections *hkl* of a tetragonal crystal are given by 1/*d*^{2} = (*h*^{2} + *k*^{2})/*a*^{2} + *l*^{2}/*c*^{2}. Since *l*^{2}/*c*^{2} is the same for both twin-related reflections, the *d* values are equal for equal *h*^{2} + *k*^{2}.

^{11}*Q* = 7*P*^{2} + *h*_{1}(*h*_{1} - 5*P*) = [7*P*^{2} + *k*_{1}(*k*_{1} + 4*P*)]/4 for coincidence condition 2*h*_{1} - *k*_{1} = 7*P*. The same holds for *h*_{2}, *k*_{2} and condition 2*h*_{2} - *k*_{2} = 7*P*.

^{12}The *d* values of reflections *hkl* of a hexagonal crystal are given by 1/*d*^{2} = (*h*^{2} + *hk* + *k*^{2})/*a*^{2} + *l*^{2}/*c*^{2}. Since *l*^{2}/*c*^{2} is the same for both twin-related reflections, the *d* values are equal for equal *h*^{2} + *hk* + *k*^{2}.

^{13}The reduced composite symmetries are the same for 3*m*1 and 31*m* *etc.*, because the twin elements *m*'(120) *etc*. are the same for both groups. For details of `structural settings', see Klapper & Hahn (2010), Appendix *A*.

^{14}Note that there are four conjugate subgroups 2/*m* of 4/*m*2/*m* of index [4]. Each of these has a single axis along one of the four body-diagonal axes of the cubic supergroup (*cf.* Müller, 2004, p. 708).

^{15}If the subgroup is chosen along one of the other body diagonals [11], [11] or [11], the types of splitting, given below, remain the same, but the Miller indices of the subforms would change.

^{16}They represent the four conjugate subgroups 2/*m* of 4/*m*2/*m*.

^{17}It is emphasized that the term `opposite' (*i.e.* related by an inversion) refers here only to the morphology of the two split forms.